Problem: Simplify the following expression: $y = \dfrac{5x^2- 24x+27}{x - 3}$
Answer: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(27)} &=& 135 \\ {a} + {b} &=& &=& {-24} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $135$ and add them together. The factors that add up to ${-24}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-9}$ and ${b}$ is ${-15}$ $ \begin{eqnarray} {ab} &=& ({-9})({-15}) &=& 135 \\ {a} + {b} &=& {-9} + {-15} &=& -24 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({5}x^2 {-9}x) + ({-15}x +{27}) $ Factor out the common factors: $ x(5x - 9) - 3(5x - 9)$ Now factor out $(5x - 9)$ $ (5x - 9)(x - 3)$ The original expression can therefore be written: $ \dfrac{(5x - 9)(x - 3)}{x - 3}$ We are dividing by $x - 3$ , so $x - 3 \neq 0$ Therefore, $x \neq 3$ This leaves us with $5x - 9; x \neq 3$.